Path integral and gaussian integral

[LayMuon]

I am trying to calculate the functional for real scalar field:

$$W[J] = \int \mathcal{D} \phi \: exp \left[{ \int \frac{d^4 p}{(2 \pi)^4}[ \frac{1}{2} \tilde{\phi}(-p) i (p^2 - m^2 +i \epsilon) \tilde{\phi}(p)} +\tilde{J}(-p) \tilde{\phi}(p)] \right]$$

Using this gaussian formula:

$$\int_{-\infty}^\infty \prod_{i=1}^N dy_i \: exp \left[ -\frac{1}{2} \sum_{i,j=1}^N y_i A_{ij} y_j + \sum_{i=1}^Ny_i z_i \right]= (2 \pi)^{N/2} (\mathrm{det} A)^{-1/2} exp \left[\frac{1}{2} \sum_{i,j=1}^N z_i (A^{-1})_{ij} z_j \right]$$

I have to discretise the p integration and then perform the integration over phi but i am unable to recover the right sign of J.

I can'r get:

$$W[J] = W[0] \: exp \left[ \frac{1}{2} \int \frac{d^4 p}{(2 \pi)^4} \tilde{J}(-p) \tilde{D}(p) \tilde{J}(p) \right]$$

Any help?

 

[LayMuon]

There was an explanation in Peskin. One should be careful with factors of 1/2. I initially messed them up.